Embedded newform 273.2.br.b.17.15

• Label: 273.2.br.b.17.15
• Level: $273$
• Weight: $2$
• Character: 273.17
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.br (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.15
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.367418 q^{2} +(-0.167841 + 1.72390i) q^{3} -1.86500 q^{4} +(-0.333187 + 0.192366i) q^{5} +(0.0616679 - 0.633392i) q^{6} +(0.931377 + 2.47640i) q^{7} +1.42007 q^{8} +(-2.94366 - 0.578683i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.367418			−0.259804			−0.129902	−	0.991527i	\(-0.541466\pi\)
							−0.129902	+	0.991527i	\(0.541466\pi\)
\(3\)	−0.167841	+	1.72390i	−0.0969033	+	0.995294i
							\(5\)	−0.333187	+	0.192366i	−0.149006	+	0.0860286i	−0.572649	−	0.819800i	\(-0.694084\pi\)
0.423643	+	0.905829i	\(0.360751\pi\)
\(7\)	0.931377	+	2.47640i	0.352027	+	0.935990i
							\(11\)	−1.86381	−	3.22821i	−0.561960	−	0.973343i	−0.997325	−	0.0730895i	\(-0.976714\pi\)
0.435365	−	0.900254i	\(-0.356619\pi\)
\(13\)	−3.56240	+	0.556129i	−0.988033	+	0.154242i
							\(17\)	−6.03970			−1.46484			−0.732421	−	0.680852i	\(-0.761610\pi\)
−0.732421	+	0.680852i	\(0.761610\pi\)
\(19\)	−2.06769	+	3.58135i	−0.474362	+	0.821618i	−0.999569	−	0.0293559i	\(-0.990654\pi\)
							0.525207	+	0.850974i	\(0.323988\pi\)
\(23\)		−	5.05239i		−	1.05350i	−0.850021	−	0.526748i	\(-0.823411\pi\)
							0.850021	−	0.526748i	\(-0.176589\pi\)
\(29\)	5.29576	+	3.05751i	0.983398	+	0.567765i	0.903294	−	0.429021i	\(-0.141141\pi\)
							0.0801039	+	0.996787i	\(0.474475\pi\)
\(31\)	−0.464104	+	0.803852i	−0.0833556	+	0.144376i	−0.904689	−	0.426072i	\(-0.859897\pi\)
							0.821334	+	0.570448i	\(0.193230\pi\)
\(37\)		−	0.549540i		−	0.0903438i	−0.998979	−	0.0451719i	\(-0.985616\pi\)
							0.998979	−	0.0451719i	\(-0.0143836\pi\)
\(41\)	9.42325	+	5.44051i	1.47166	+	0.849666i	0.999493	−	0.0318414i	\(-0.0101371\pi\)
							0.472171	+	0.881507i	\(0.343470\pi\)
\(43\)	1.64275	+	2.84533i	0.250517	+	0.433908i	0.963668	−	0.267102i	\(-0.0860660\pi\)
							−0.713151	+	0.701010i	\(0.752733\pi\)
\(47\)	−7.98492	+	4.61010i	−1.16472	+	0.672452i	−0.952431	−	0.304755i	\(-0.901425\pi\)
							−0.212290	+	0.977207i	\(0.568092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.17.15	yes	64
3.2	odd	2	inner	273.2.br.b.17.18	yes	64
7.5	odd	6		273.2.y.b.173.18	yes	64
13.10	even	6		273.2.y.b.101.15	✓	64
21.5	even	6		273.2.y.b.173.15	yes	64
39.23	odd	6		273.2.y.b.101.18	yes	64
91.75	odd	6	inner	273.2.br.b.257.18	yes	64
273.257	even	6	inner	273.2.br.b.257.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.15	✓	64	13.10	even	6
273.2.y.b.101.18	yes	64	39.23	odd	6
273.2.y.b.173.15	yes	64	21.5	even	6
273.2.y.b.173.18	yes	64	7.5	odd	6
273.2.br.b.17.15	yes	64	1.1	even	1	trivial
273.2.br.b.17.18	yes	64	3.2	odd	2	inner
273.2.br.b.257.15	yes	64	273.257	even	6	inner
273.2.br.b.257.18	yes	64	91.75	odd	6	inner